The integral Si(x)= 0 sin t -dt, called the sine-integral function, has important applications in optics.

(A) Graph the function fx) = x-2 sinx in the viewing rectangle [0, 8] by [-2, 8]. (B) On which interval is the average rate of change larger: [1, 2] or [2, 3]? (C) At which value of x is the instantaneous rate of change larger: x = 2 or x = 5? (D) Check your visual estimates in part(c) by computing f'（x) and comparing the numerical values ​​of f'（2）and f'（5）

USE THE SUM FORMULA

(a) Find and locate local extreme values off(x) = (x+ 1)^4−4x−2, (b) Then find the intervals of increase, decrease and concave upward, concavedownward and inflection points.

Expand tan(x) in powers of x up to first six terms and estimate the value of the function with x =30°.

Find the equations of both of the tangent lines to the ellipse x² + 4y² = 36 that pass through the point (12, 3). List the line with the smaller slope first. ソミ

tanh (1.3288x )using bisection Determine the real root of f(x)=L.1288 method to locate the root. Employ initial guesses of x = 1.000 and x = 2.000 and iterate until the root is obtained to two decimal places. Express f(x,)J; f(x,) or f[(x, ),]· f(x,)to five decimal places, and the rest to three decimal places in every iteration.

The central difference approximation to the derivative f(x) = x² sin(2x) at x = π/3 is: (use step size of 0.005) Type your answer...

QA: (inspired by a problem from Stewart's textbook) Suppose that there is some function f whose _derivative_ is f'=sin(x)/x, with f '(0)=1 by definition rather than DNE. Draw that, on the interval [-4pi,+4pi]. (i) On what intervals is the original f increasing? Decreasing? Indicate the intervals on the graph as well as writing them in interval notation like [0,pi] (ii) At what x values does f have a local max? A local min? Indicate them on the graph as well as writing them out like: maxes at ...; mins at .... (iii) On what intervals is f CD? CU? Indicate the intervals on the graph as well as writing them in interval notation. You won't be able to find the exact values using formulas; approximate values are fine. (iv) At what x values does f have an inflection point? Indicate them on the graph as well as writing them out like: IP at approximate values are fine, since it's impossible to find exact values] (v) Sketch a graph of f, starting at f(0)=0. [again,

The backward difference approximation to the derivative f(x) = x² sin (2x) at x = π₁ π/3 is: (use step size of 0.005) Type your answer...